Amplitude and Power

When you have two random signals:

S1: 3 4 6 8 3 5 7 9 2 4 6 8 3 6 9

S2: 4 13 21 45 32 8 7 9 8 3 5 4 6 5 7

Standard deviation S1 = 2.3258383

Standard deviation S2 = 11.9654264

When you add S1 and S2 together you get signal S3

S1 + S2 = S3: 7 17 27 53 35 13 14 18 10 7 11 12 9 11 16

Standard deviation S3 = 12.4077549

When you add the standard deviation S1 to the standard deviation S2

2.3258383 + 11.9654264 = 14.2912647

So it turns out that the added standard deviation of S1 + S2 is uneven with the standard deviation of S3.

Why is this so?

This has to do with the physicality of signals. When you add two random signals together, the combined random signal is not an addition of the amplitudes but of the Power. In electronics, the Power = U²/ R (U is the amplitude here)

So we have to square the signals first:

S1: 9 16 36 64 9 25 49 81 4 16 36 64 9 36 81

S2: 16 169 441 2025 1024 64 49 81 64 9 25 16 36 25 49

Standard deviation S1: 26.615427

Standard deviation S2: 552,791212

S1 + S2 = S3: 25 185 477 2089 1033 89 98 162 68 25 61 80 45 61 130

Standard deviation S3 = 579.406639

Now the standard deviation of the combined signal is equal to the added standard deviation of the individual signals;

26.615427 + 552.791212 = 579.406639

The above is also the reason that when calculating the standard deviation, the deviation is squared with the average, with this you actually calculate the Power of the deviation instead of just the deviation.